As shown in a series of publications that led to the present work[1]}, [2]}, [3]}, [4]}, [5]}, [6]} - it is possible to take into account the non-equilibrium nature of the electric screening in the case of a ideal Hall bar on the basis of the least dissipation principle [7]}, [8]}, [9]}. The stationary state can then be deduced from the minimization of the power under the global constraints, instead of solving independently the drift-diffusion equations. This approach has been applied successfully in the case of the classical Hall effect without spin [4]}, [5]}, [6]}. It has been shown that the charges accumulated at the edges are not static but generate a non-uniform longitudinal current \(\delta J_x(y)\) , which has not been derived before. Physically, this surface current is responsible for the well-known robustness of the Hall voltage, which can be measured while using a good or a bad voltmeter (i.e. with huge variations of the electric leakage at the edges [13]}), because the electric charges accumulated are renewed permanently despite the zero transverse current. Furthermore, if a secondary passive circuit is contacted the two edges of the Hall bar, the variational method shows that the electric current injected is mainly carried by the longitudinal surface currents \(\delta J_x(y)\) , instead of the transverse current \(J_y\)  [6]}.
